• 沒有找到結果。

# New Gauge Symmetries from String Theory

N/A
N/A
Protected

Share "New Gauge Symmetries from String Theory"

Copied!
21
0
0

(1)

### New Gauge Symmetries from String Theory

Pei-Ming Ho

Physics Department

National Taiwan University

Sep. 23, 2011@CQSE

(2)

### Gauge vs Global

For covariant quantities

Old definitions in some textbooks:

U = independent of spacetime  global U = function of spacetime  gauge (local)

(3)

Gauge vs Global (2)

• Translation symmetry is usually considered as a global symmetry.

• Different interpretation of the transformation:

Translation by a specific length L can be

“gauged” (defined as a gauge symmetry)

 equivalence:

(compactification on a circle).

• Gauge potential is useful but not necessary.

x(t) » x(t) + L

(4)

Gauge vs Global (3)

• 2nd example:

base space = R+ with Neumann B.C. at x = 0.

• 3rd example:

space  an interval of length L/2 w.

Neumann B.C.

Space/(subgroups of isometry)  orbifolds

x® - x, f(x) ® f(-x)

(5)

Gauge vs Global (4)

New (better) definition:

• Gauge Symmetry

– Transformation does not change physical states.

– A physical state has multiple descriptions.

– Transformation changes descriptions.

• Global Symmetry

– Transformation changes physical states.

This is a more fundamental distinction than spacetime-dependence.

(6)

(1)

(0)

(1)

(0)

(2)

(1)

(1)

(1)

(2)

(2)

(0)

(7)

### Non-Commutative Gauge Theory

• D-brane world-volume theory in B-field background. [Chu-Ho 99, Seiberg-Witten 99]

• Commutation relation determined by B-field.

• U(1) gauge theory is non-Abelian.

[xi, xj] = iQij

(8)

### Lie 3-Algebra Gauge Symmetry

• BLG model for multiple M2-branes [07].

• needed for manifest compatibility with SUSY.

(ABJM model does not have manifest full SUSY.)

[Ta, Tb, Tc] = f abcdTd A = Aiab(Ta Ä Tb)dxi

(Dif)a = ifa + Aibcfd f bcda

(9)

### Generalization of Lie 3-algebra

• All Lie n-algebra gauge symmetries are special cases of ordinary (Lie algebra) non-Abelian

gauge symmetries.

[Ta1 , Ta2 ,, Tan] = f a1a2anan+1Tan+1 A = Aa1a2an-1 (Ta1 Ä Ta2 ÄÄ Tan-1 ) f = faTa

[A, f] = Aa1a2an-1fa

n f a1a2anan+1Tan+1

(10)

### More Gauge Theories

• Abelian Higher-form gauge theories

• Self-dual gauge theories

• Non-Abelian higher-form gauge theories

• NA SD HF GT w. Lie 3 (on NC space?)

A(n) = n!1 Ai1i2indxi1 Ùdxi2 ÙÙdxin F(n+1) = dA(n)

dA(n) = dL(n-1) dF(n+1) = 0

(11)

### Dirac Monopole

• Jacobi identity (associativity)

is violated in the presence of magnetic monopoles.

• Distribution of magnetic monopoles  gauge bundle does not exist 

2-form gauge potential (3-form field strength) (Abelian bundle  Abelian gerbe)

Q: How to generalize to non-Abelian gauge theory?

[[Di, Dj ], Dk] +[[Dj, Dk], Di] +[[Dk, Di], Dj] = 0

(12)

### Self-Dual Gauge Fields

• Examples:

type IIB supergravity type IIB superstring M5-brane theory twistor theory

• D=4k+2 (4k) for Minkowski (Euclidean) spacetime.

(13)

### Self-Dual Gauge Fields

• When D = 2n, the self-duality condition is

• a.k.a. chiral gauge bosons.

• How to produce 1st order diff. eq. from action?

F = *F

F = n!1 Fi1i2in dxi1 Ù dxi2 ÙÙ dxin

*F = n!1 (D-n)!1 e i1i2iD F i1i2in dxin+1 Ù dxin+2 ÙÙ dxiD

(14)

### Chiral Boson in 2D

[Floreanini-Jackiw ’87]

• Self-duality

• Lagrangian

• Gauge symmetry

• Euler-Lagrange eq.

• Gauge transformation

• 1-1 map btw space of sol’s and chiral config’s

### f  ˙ - ¢  f  = 0

L = 12 f ¢ ( ˙ f - ¢ f )

x(t -x)f = 0

Þ (t -x)f = g(t)

(15)

• Due to gauge symmetry, f  is not an observable.

k=f is an observable  the Lagrangian is

• Formal nonlocality is unavoidable, although physics is local.

• Equivalent to a Weyl spinor in 2D in terms of the density of solitons.

• Lorentz symmetry is hidden.

L = 12 k(t, x)

## ( ò

dye(x- y) ˙ k (t, y) - k(t, x)

## )

[

(t, x),

(t, y)] = i

### e

(x- y)

(16)

• No vector potential needed for gauge symmetry Vector potential is useful for defining

cov. quantities from deriv. of cov. quantities.

But it is not absolutely necessary.

• The formulation can be generalized to self-dual higher-form gauge theories.

Dm = m + Am

(17)

Example: M5-brane theory (D=5+1)

[Howe-Sezgin 97, Pasti-Sorokin-Tonin 97, Aganagic-Part- Popescu-Schwarz 97, …]

[Chen-Ho 10]

[Ho-Imamura-Matsuo 08]

(18)

### Questions

• What is the geometry of Abelian higher-form gauge symmetry?

(bundles  gerbes?)

• How to define non-Abelian higher-form gauge symmetry?

• Can we generalize the notion of covariant derivative and field strength?

• Do we still need the covariant derivative?

(19)

### Non-Abelian Self-Dual 2-Form Gauge Potential

• M5-brane in the (3-form) C-field background.

[Ho-Matsuo 08, Ho-Imamura-Matsuo-Shiba 08]

• Non-Abelian gauge symmetry for 2-form gauge potential

• Nambu-Poisson algebra (ex. of Lie 3-algebra) (Volume-Preserving Diffeomorphism)

• Part of the gauge potential  VPD

• 1 gauge potential for 2 gauge symmetries!

[Ho-Yeh 11]

(20)

### Self-Dual 2-Form Gauge Field

• Need non-locality to circumvent no-go thms for multiple M5-branes. [Ho-Huang-Matsuo 11]

• In “5+1” formulation, decompose all fields

into zero modes vs. non-zero modes in the 5-th dimension.

• Zero modes  1-form potential A in 5D

• Non-zero modes  2-form potential B in 5D

(21)

• Self-duality: Instantons, Penrose’s twistor theory applied to Maxwell and GR.

• In 4D, 2-form ≈ 0-form, 3-form ≈ (-1)-form, 4-form ≈ trivial through EM duality.

• Expect more new gauge symmetries from string theory to be discovered.

• “Symmetry dictates interaction” -- C. N. Yang.

generalization of Weyl’s gauge invariance, to a possible new theory of interactions

Especially, the additional gauge symmetry are needed in order to have first order differential equations as equations

We further want to be able to embed our KK GUTs in string theory, as higher dimensional gauge theories are highly non-renormalisable.. This works beautifully in the heterotic

another direction of world volume appears and resulting theory becomes (1+5)D Moreover, in this case, we can read the string coupling from the gauge field and this enables us to

We define Flat Direction Hybrid Inflation (FDHI) models as those motivated by the properties of moduli fields or flat directions of the standard model. For moduli fields with no

● moduli matrix ・・・ torus action around the fixed points. vortex

QCD Soft Wall Model for the scalar scalar &amp; &amp; vector vector glueballs glueballs

• Emergent Z_k 1-form &amp; 2-form symmetry. BF theory